The Geometry of Minkowski Spaces A Survey. Part II

Transcription

1 The Geometry of Minkowski Spaces A Survey. Part II H. Martini, Fakultät für Mathematik, Technische Universität Chemnitz, D Chemnitz, Germany. martini@mathematik.tu-chemnitz.de K. J. Swanepoel Department of Mathematics, Applied Mathematics and Astronomy, University of South Africa, PO Box 392, Pretoria 0003, South Africa. swanekj@unisa.ac.za MSC 2000 Subject Classification: Primary 46-02; secondary 52A21, 46B20. 1

2 Abstract. In this second part of a series of surveys on the geometry of finite dimensional Banach spaces (Minkowski spaces) we discuss results that refer to the following three topics: bodies of constant Minkowski width, generalized convexity notions that are important for Minkowski spaces, and bisectors as well as Voronoi diagrams in Minkowski spaces. Keywords: Minkowski Geometry, Minkowski spaces, finite dimensional normed spaces, bisectors, Voronoi diagrams, bodies of constant width, generalized convexity 1 Introduction This paper is the second part of a planned series of surveys on Minkowski Geometry, which is the geometry of finite dimensional normed linear spaces (= Minkowski spaces). The origins and basic developments of Minkowski Geometry are connected with names such as Riemann, Minkowski and Busemann, see the Preface of [283] and [196, 2] for more information. This field can be located at the intersection of Finsler Geometry, Banach Space Theory and Convex Geometry, but it is also closely related to Distance Geometry (in the spirit of Menger and Blumenthal [28]) and was enriched by many results from applied disciplines such as Operations Research, Optimization, Theoretical Computer Science and Location Theory. The motivation for this series of surveys is given by the facts that there are hundreds of papers in Minkowski Geometry, widespread in very different fields, previous surveys are old, and the recent excellent monograph [283] covers mainly the analytical part of the theory. In Part I [196] we placed special emphasis on planar results, in some cases with simplified proofs. Thus Part I can be seen as fundamental for the parts following it. In Part II we survey three topics from Minkowski Geometry that are very geometric in nature and show interesting relations to further disciplines, such as Classical Convexity, Abstract Convexity, Computational Geometry and the Foundations of Geometry. Bodies of constant width are well studied in Convex Geometry. The extensive knowledge on these special convex bodies in Euclidean space is summarized in the surveys [60] and [131]. Although these surveys and the monograph [283] also contain material on bodies of constant width in Minkowski spaces, a complete summary is missing. Our first 2

3 section below is the first and, as we believe, complete survey on this subject. It gives an example of how strongly Classical Convexity and Minkowski Geometry are related to each other. Generalized convexity notions in the sense of metric convexity were studied in Distance Geometry and Abstract Convexity. Some of these notions are especially interesting in the context of Minkowski Geometry and applications thereof, e.g. in Location Science and Computational Geometry. We present an overview, particularly showing the state of the art regarding the theory of d-convex sets. We also take into consideration other types of generalized convexity notions in normed linear spaces. Geometric properties of bisectors in Minkowski spaces yield various deep characterizations of special normed linear spaces, or of Minkowski spaces within more general classes of spaces. This viewpoint is also related to Foundations of Geometry. Furthermore, the study of Voronoi diagrams is based in a natural manner on the geometry of bisectors and their extensions. During the last decade, these topics were mainly investigated in Computational Geometry, in many cases even for linear spaces equipped with a nonsymmetric unit ball. Since this more general approach is still based on the tools typical for Minkowski Geometry, it is also considered in the third part of this survey. Finally we mention here the contents of our planned Part III which is in preparation: smoothness of norms, Chebyshev sets, isometries (Hyers-Ulam and Beckman-Quarles type theorems, as well as Banach-Mazur distance), isoperimetric problems, and various notions of orthogonality and angle measures. A Part IV is also planned and will contain subjects more related to Discrete Mathematics. We finish this introduction by some notation used in all three sections below. We let M d denote a d-dimensional Minkowski space, i.e., a d-dimensional real normed linear space with norm and unit ball B := {x M d : x 1}. As usual we write E d for the d-dimensional Euclidean space. Points and vectors are in boldface. The origin is denoted by o. A convex body K M d is a compact, convex subset of M d with nonempty interior. For two distinct points x, y M d we denote the linear segment joining them by [x, y], and the ray emanating from x and passing through y by [x, y. (This notation for a segment differs from the notation in Part I. This is so that it corresponds to the usual notation for d-segments, see the section on generalized convexity notions below). For further general notation and definitions we refer to Part I [196]. 2 Bodies of constant width in Minkowski spaces 2.1 Introduction If the distance between any pair of parallel supporting hyperplanes of a convex body K in d-dimensional Euclidean space E d, d 2, is the same, then K is called a body of constant width. Since the time of Euler or even further back it has been known that there are many non-spherical bodies of constant width, the most famous one being the Reuleaux triangle, cf. [232]. Most of the results on bodies of constant width derived up to 1934 are collected in the classical monograph [40]. More recent surveys, showing the pertinent results obtained before 1993, are [60] and [131, Section 5]. Various related concepts (like bodies of constant brightness, etc.) are also discussed in the books [98] and [244]. And there is even a 3

4 monograph on bodies of constant width, see [208]. It is natural to extend the notion of bodies of constant width to Minkowski spaces, and in [60] and [131] this more general point of view is also taken into consideration. In addition, we refer to Chapter 4 of Thompson s monograph [283] and [36, Chapter V], where wider discussions of some geometric properties of bodies of constant width in Minkowski spaces are given. However, with many references not cited in [60], [131] and [283], and since the literature on such bodies in Minkowski spaces is sufficiently grown and widespread, one is motivated enough to write an independent summary on this topic. Nevertheless, for practical reasons our sequence of subsections will follow that from [60] and [131]. 2.2 Basic Notions We start by introducing Minkowski analogues of various notions from (Euclidean) convex geometry. Note that a unit functional ϕ in the dual (M d ) of M d is associated in a natural way with a directed hyperplane of M d. The Minkowski support function of a convex body K is defined by h(k, ϕ) := sup{ϕ(x) : x K} for any unit functional ϕ (M d ). Thus h(k, ϕ) is the signed distance from the origin o to the supporting hyperplane H = {x : ϕ(x) = 1} of K. The Minkowski width (or Minkowski breadth) of K in direction ϕ is given by w(k, ϕ) = h(k, ϕ) + h(k, ϕ), and K is said to be of constant Minkowski width w(k) R + if w(k, ϕ) = w(k) for any unit functional ϕ. For later use, we also introduce the central symmetral K of a convex body K M d, defined by K := 1 (K + ( K)), 2 where the (Minkowski or) vector addition is defined by K 1 + K 2 := {x + y : x K 1, y K 2 }. A segment [p, q], whose different endpoints p, q are from the boundary bd K of K, is called a diametrical chord (or affine diameter) of K if there exist two parallel supporting hyperplanes H 1 H 2 of K such that p H 1 and q H 2 ; in this situation we say that [p, q] is generated by H 1, H 2. In Euclidean space E d, a chord [p, q] of K is called a normal of K at p bd K if K has a supporting hyperplane H 1 with p H 1 such that [p, q] is orthogonal to H 1. If, moreover, there is a supporting hyperplane H 2 q which is parallel to H 1, then [p, q] is called a double normal of K. To transfer these notions to Minkowski spaces, we first need a suitable notion of orthogonality. Let H be a hyperplane and u o a vector in M d. The vector u (or a line with u as its direction vector) is said to be normal to H, denoted by u H, if the two supporting hyperplanes of the unit ball B which are parallel to H generate a diametrical chord of B having direction u and passing through o. (Thus, if B is not strictly convex there may be infinitely many directions normal to H.) A chord [p, q] of a convex body K M d is a Minkowski normal of K at p bd K if K has a supporting hyperplane H 1 p such that [p, q] is normal to H 1. Also, [p, q] is a Minkowski double normal of K if, in addition, there is a supporting hyperplane H 2 q parallel to H 1. Finally, a point x from the interior int K of K M d is called an equichordal point of K if all chords of K passing through x have the same Minkowski length. 2.3 Geometric properties and characterizations There are various well known properties of bodies of constant width in E d, d 2, characterizing them within the class of all d-dimensional convex bodies. A collection of such 4

5 characterizations is given in [60, Section 2], and the analogous characterizations of bodies of constant Minkowski width can be summarized by Theorem 1. A convex body K M d is of constant Minkowski width w(k) R + if and only if one of the following statements holds true. (1) The central symmetral K satisfies K = w(k) 2 B. (2) For each pair H 1, H 2 of parallel supporting hyperplanes of K and every direction u normal to H 1, H 2 there exists a diametrical chord of K having direction u and generated by H 1, H 2. (3) For each pair H 1, H 2 of parallel supporting hyperplanes of K, every diametrical chord of K generated by H 1, H 2 is normal to H 1, H 2. (4) All diametrical chords of K have length w(k). Proofs of the equivalences of (1) and (2) to constant Minkowski width are presented in [85], and the analogues referring to (3) and (4) are verified in [60, Section 2], see also [88] and [127]. Our next statement (I) is also proved in [85], and (II) is an easy consequence of it. In both (I) and (II) there are now restrictions on the unit ball. Theorem 2. In a Minkowski space M d with smooth and strictly convex unit ball the following statements hold true. (I) A convex body K M d is of constant Minkowski width if and only if any two parallel Minkowski normals of K coincide. (II) A convex body K M d is of constant Minkowski width if and only if every chord [p, q], which is a Minkowski normal of K at p, is also a Minkowski normal of K at q. It should be noticed that without any restrictions on B the above coincidences of Minkowski normals still imply that K is of constant Minkowski width, but the converse may fail to be true. Vrećica [298] has shown that a convex body K M d is of constant Minkowski width if and only if for all x, y int K there is a set C of constant Minkowski width such that C int K and x, y bd C. In [189] it is proved that a convex body K in the Euclidean plane is of constant (Euclidean) width w(k) = diam K if and only if any two mutually perpendicular chords of K with a common point have total length not smaller than diam K. One might ask for a characterization of those norms where the analogous statement holds, if Euclidean perpendicularity of the chords is replaced by normality. The formulation of the monotonicity lemma in Minkowski spaces involves only Minkowski balls, cf. [196, Section 3.5] or [283, Lemma 4.1.2]. In [132] Heppes proves a characterization theorem which can be treated as a monotonicity lemma for bodies of constant width in E 2. Grünbaum and Kelly [123] extend one of the implications from [132] to strictly convex Minkowski planes, and in [19] the characterization theorem of Heppes is extended to arbitrary Minkowski planes, namely as the two-dimensional part of the following 5

6 Theorem 3. Any hyperplane section S of a convex body K M d of constant Minkowski width splits K into two compact, convex sets such that at least one of them has the same diameter as S. For d = 2 this property characterizes the bodies of constant Minkowski width within the class of two-dimensional convex bodies. Similar characterization theorems, referring to double normals and the unimodality of chord parametrizations of planar curves, are obtained in [16] and [17]. A further characterization of bodies of constant Minkowski width is given in [134]. Another type of result is the characterization of special representatives within the class of bodies of constant Minkowski width. For example, Petty and Crotty [220] prove Theorem 4. If a convex body K M d has constant Minkowski width and, in addition, an equichordal point, then K is homothetic to the unit ball B. In a different way, a part of their proof was earlier obtained by Hammer [127], together with further observations on diametrical chords in Minkowski planes. Hammer and Smith [128] prove that if all binormal chords of a curve of constant width in the Euclidean plane divide its circumference into two equal parts, then it is a circular disc. The Minkowski version of this theorem is announced there without proof. Let r denote the inradius of a convex body K M d, i.e., the largest real number such that K contains some translate of rb, and R be the circumradius of K, that is the smallest real number such that K is contained in some translate of RB. The boundaries of these translates of rb and RB are said to be inspheres and circumspheres of K, respectively. Chakerian [57] (see also [88]) shows that the following statement holds. Theorem 5. If a convex body K M d has constant Minkowski width w R +, then the equality r + R = w holds. Moreover, the corresponding insphere and circumsphere of K are concentric. Conversely, these properties do not imply constant width in M d, but a generalization of Theorem 5 is given in [240]; cf. Section 2.6 below. Some geometric properties of planar bodies of constant width in the Minkowski plane (M 2 ), whose unit ball is the isoperimetrix I (i.e., the convex figure of minimal perimeter for given area) of the original plane M 2, are discussed in [150] and [55]. For example, if C is a figure of constant width w R + relative to (M 2 ), then A(C) + A(C, C) = w 2 A(I), where A(C) is the area of C and A(C, C) denotes the mixed area (cf. [244, 5.1]) of C and C. They also give further relationships between M 2 and (M 2 ), e.g.: 1. if C has constant width w relative to (M 2 ) and P (C) is the perimeter of C relative to M 2, then P (C) = w A(I), 2. if C is a smooth convex curve of constant width relative to (M 2 ) such that each of its diametrical chords bisects the Minkowski circumference (or the Minkowski area), then C is homothetic to I. 6

7 2.4 Special bodies of constant width The simplest figure of constant width w R + in E 2 apart from a circle is the Reuleaux triangle whose first mechanical usage is ascribed by Reuleaux to Hornblower, the inventor of the compound steam-engine, see [232, 155]. It is bounded by three circular arcs of radius w which are centred at the vertices of an equilateral triangle (with side-length w). There are different ways to define its analogue for Minkowski planes. Replacing the notions circular and equilateral by their Minkowski analogues, Ohmann [211], Chakerian [56] and Wernicke [300] describe Minkowski Reuleaux triangles as extremal figures with respect to certain metrical problems, see also Section 2.8 below. In addition, the planes with parallelograms or centrally symmetric hexagons as unit circles are the only ones where Reuleaux triangles exist as Minkowski circles, cf. [300]. The paper [231] of Reimann is also geometrically related to Reuleaux triangles in Minkowski planes. Sallee [237] extends these considerations by constructing certain Minkowski Reuleaux polygons. Another type of Reuleaux polygon in M 2 has been described by Hammer [127]. Its Euclidean variant yields figures of constant width bounded by finitely many circular arcs of possibly different radii (see Rademacher and Toeplitz [228, p. 167] for the Euclidean case). Petty [216] uses the Minkowski analogue of the notion evolute to construct special curves of constant width in Minkowski planes. 2.5 Completeness in Minkowski spaces A bounded subset C of M d is called complete (or diametrically complete or diametrically maximal) if it cannot be enlarged without increasing its diameter diam C := sup x,y C x y. In other words, C is complete if it is the intersection of all translates x + λb with x C and λ diam C. Any complete set C having the same diameter diam C = diam K as a convex body K M d with K C is said to be a completion of K. Various results on complete sets in Minkowski spaces are compiled in [36, Chapter V] and [112]. In Euclidean space, the following two statements hold true. A convex body K E d is complete if and only if it is of constant width. Every convex body K E d has at least one completion. The first statement is the classical theorem of Meissner, see [204] and [40, 64], and the second one is known as Pál s theorem, cf. [213] and [40, 64]. It is easy to see that constant width implies completeness also in Minkowski spaces, see, e.g., [88, 2] for an explicit proof. For d = 2, 3 and smooth unit balls Meissner [204] investigated the converse, followed by Kelly [149] without restrictions regarding the dimension and smoothness. However, the proofs in [204] and [149] with respect to this converse implication are erroneous for d 3: there are complete sets that are not of constant width, even if the unit ball is smooth and strictly convex. Such an example was constructed by Eggleston [85], who also gives an example with a polyhedral unit ball. Thus, the extension of Meissner s classical theorem to Minkowski spaces is true only for d = 2. However, in higher dimensions the completeness of K M d implies constant width if K is smooth and strictly convex, cf. [88, Theorem (3.18)]. However, there is a related equivalence theorem holding for all Minkowski spaces. A convex body K M d is said to have the spherical 7

8 intersection property if K is the intersection of all balls with centre x K and radius diam K, cf. also Section 2.6. Using this notion Eggleston [85] also proves Theorem 6. A compact set X M d is complete if and only if it has the spherical intersection property, or if and only if each boundary point of X has distance diam X from at least one other point of X. An alternative proof of the first statement in this theorem can be found in [88, 3]. Eggleston [85] compares Minkowski spaces in which the classes of complete sets, sets of constant width and spheres coincide (cf. also [88, 3]). In [84] he erroneously states that the only Minkowski spaces in which sets of constant width are necessarily balls are those whose unit ball B is a parallelotope. In order to obtain the right answer to this question, one has to look at the classification of all irreducible sets, i.e., of all convex bodies K centred at the origin for which the identity K = Q + ( Q) is only possible if Q is centrally symmetric, see Grünbaum [121] and Shephard [249] for first investigations in this direction. A thorough study of irreducible polytopes (i.e., polyhedral unit balls for which sets of constant Minkowski width are necessarily balls) is given by Yost [304], see also [244, 3.3]. For example, each convex d-polytope, d 3, centred at the origin and having less than 4d vertices is irreducible. Investigating four-dimensional polyhedral unit balls in view of irreducibility, Payá and Yost [215] obtain examples of Minkowski spaces that are interesting regarding the n-ball property and semi-m-ideals (notions from Banach space theory). Eggleston [85] also shows that if the unit ball B is a d-dimensional parallelotope, then every complete set is homothetic to the unit ball. The converse is proved by V. Soltan [261], i.e., we have Theorem 7. If every complete set in M d is homothetic to the unit ball B, then B has to be a parallelotope. The two-dimensional case is contained in Theorem 8.1 of [127]. The infinite dimensional case has been looked at by Dalla and Tamvakis [67] and Franchetti [90]; they also investigate completeness in relation to constant width. Eggleston [85] also shows the following Theorem 8. Any bounded set S M d can be embedded in a complete set of the same diameter. Uniqueness of completion in Minkowski planes plays an essential role with respect to Borsuk s partition conjecture in such planes, see [36, 33] and Section 2.10 below. Groemer [112] investigates the uniqueness of completion in d-dimensional Euclidean and Minkowski spaces, e.g. in view of maximal tight covers of a bounded set S M d (which are convex bodies K of maximal volume satisfying S K and having the same diameter as S). He shows that every maximal tight cover of a bounded set S M d is a completion of S and that any two such completions are translation equivalent; for strictly convex norms there is precisely one completion of this type. Furthermore, symmetry properties of such maximal completions are studied, and cardinalities of the set of completions for a given set (without the maximality assumption) are discussed. Various results from [112] are new even for the Euclidean case. We also refer to [21], where some of Groemer s 8

9 theorems are summarized and used to get further results. Vrećica [298] observes that each bounded subset of M d has a completion within any circumball (i.e., a smallest Minkowski ball containing this set); a sharpening is obtained in [88]. Sallee [242] describes methods for constructing complete sets in M d containing an arbitrarily given set X. He pays special attention to the problem of preassigning boundary parts of such sets and, following ideas from [239], also to questions of preserving symmetries (if this is desired). Some of Sallee s results are reproved in [21], and it is shown there that these constructions carry over to infinite dimensions, as the result of Vrećica [298] does. On the other hand, it is shown in [21] that, while in finite dimensions there is a completion C satisfying C B = S B, the analogue fails in infinite dimensions (here B denotes a circumsphere of the bounded set S M d ). 2.6 Intersection properties Due to Eggleston [85] a convex body K M d has the spherical intersection property if K is the intersection of all balls with centre x K and radius diam K (see also Theorem 6 above). In Euclidean space, the property of constant width and the spherical intersection property are equivalent, cf. [85] and [60, p. 62]. In Minkowski spaces this is no longer true, but the spherical intersection property is still equivalent to the notion of completeness, see Theorem 6 above and [85]. For a body of constant width K M d let t(k) denote the smallest number of balls whose intersection equals K. V. Soltan [263] proves some results on the possible values of t(k). For example, he derives necessary and sufficient conditions for t(k) <. For d = 2, he determines unit balls that admit given even values for t(k). Groemer [112] uses the spherical intersection property to study bounded sets with unique completion in M d. Sallee [240] generalizes Theorem 5 above. He shows that it still holds if one replaces constant width w R + by the spherical intersection property. Section 2 of [21] contains also some properties of sets which can be presented as intersections of balls. Analogously, Bavaud [22] introduces the s-adjoint transform in E 2, which associates to a two-dimensional convex body K its s-adjoint K (s) as the intersection of all discs of radius s whose centres are from K. Explaining the connection of this transform to hyperconvexity (cf. our section on d-convexity as well as [25, 52, 198]) and to the concept of constant width, he gives a new construction of the completion of a compact set in the plane. He also presents applications related to stochastic point processes and statistical mechanics. One might ask for extensions of these Euclidean results to Minkowski planes. Baronti and Papini [21] establish various properties connecting completeness and the spherical intersection property for infinite dimensions. Kupitz and Martini [163] consider a related notion: a set S E 2 of diameter 1, say, has the weak circular intersection property if the intersection of all unit discs whose centres are from S is a set of constant width 1. They determine the least number of points to be added to a finite planar set of given diameter such that the resulting set has the weak circular intersection property. Nothing seems to be known about this notion in higher dimensions or Minkowski spaces. The dissertation [186] contains Helly-type theorems for sets of constant Minkowski width. 9

10 2.7 Curvature and mixed volumes Let K E d be a body of constant width w(k) R + whose boundary bd K is twice continuously differentiable, and let R 1 (u),..., R d 1 (u) denote the principal radii of curvature at the point x bd K with outward unit normal u. If F 1 (u) := d 1 i=1 R i(u), then F 1 (u) + F 1 ( u) = (d 1) w(k) (1) for each u S d 1, see [40, p. 128]. This can be extended to all bodies of constant width by introducing the surface area function S(K, ω), ω B, where B denotes the field of Borel subsets of S d 1. The mixed surface area function S(K 1,..., K n 1 ; ω) of convex bodies K 1,..., K n 1 E d is a completely additive set function defined for ω B. Thus for any convex body K 0 E d the mixed volume (cf. [244, 5.1]) is given by V (K 0, K 1,..., K n 1 ) = 1 n S d 1 h(k 0, u)s(k 1,..., K n 1; du). (2) Here the presented integral is the Radon-Stieltjes integral of h(k 0, u) with respect to the function S(K 1,..., K n 1 ; ω). Then S(K, ω) := S(K,..., K, ω) is called the surface area function (or first curvature measure of second kind), cf. [244, 4.2]. Since, if K is of constant width, K + ( K) is a homothet of the Euclidean ball, the linearity and homogenity of S(K, ω) as a function of K imply S(K, ω) + S(K, ω) = w(k) µ(ω), (3) where µ(ω) is the spherical Lebesgue measure of the Borel set ω S d 1. By a known integral representation of S(K, ω) in terms of F 1 (see [244, 4.2]), (3) is equivalent to (2) if K has a sufficiently smooth boundary, and basically due to the Aleksandrov-Fenchel- Jessen uniqueness theorem (cf. [244, 7.2]) it can be concluded that (3) is a characteristic property of all bodies of constant width in E d. It turns out that, in terms of relative differential geometry (see also [40, 38] and [60, 6]), these relations can be extended to Minkowski spaces. Namely, for a smooth convex body K E d let L(bd K, x) denote the canonical linear mapping of its tangent space bd K x at x bd K into the tangent space of S d 1 at u S d 1, where x and u are connected by the usual Gauss mapping via parallel normals, with u as unit outward normal of bd K at x. If for a smooth unit ball B in M d, B e denotes the tangent space of bd B at e having the same unit normal as bd K at x, then the linear mapping J : bd K x B e can be defined by J = L(bd K, x) L 1 (bd B, e) and is, as the canonical linear mapping of bd K into bd B via parallel normals, invertible with J 1 = L(bd B, e) L 1 (bd K, x). Thus we may write J = J(u), where u is the outward unit normal of bd K at x, and of bd B at e. The relative principal radii of curvature R 1,..., R d 1 of bd K at x are the reciprocals of the eigenvalues of J(u), and the corresponding relative principal directions are the eigenvectors of J(u). Following [40, p. 64], we write {R 1,..., R ν } for the νth elementary symmetric function of the relative principal radii of curvature R 1,..., R d 1, and we let F ν (K, u) = {R 1,..., R ν } at u as above. With this notation, Chakerian [57] proves 10

11 Theorem 9. Let K be a smooth convex body of constant Minkowski width w(k) R + in M d. Then R i (u) + R d i ( u) = w(k), u S d 1, i = 1,..., d 1, and, in particular, F 1 (K, u) + F 1 (K, u) = (d 1) w(k), u S d 1. We set S(K,..., K, B,..., B; ω) =: S(K, r; B; ω) by using the notation given in connection with (2). Here the convex body K appears r times and B appears n r 1 times. Chakerian [57] also proves Theorem 10. If K is a body of constant Minkowski width in M d, then S(K, 1; B; ω) + S(K, 1; B; ω) = 2 S(B, ω) (4) with ω B. Conversely, if the unit ball B is smooth, (4) implies that K is of constant Minkowski width. Here the smoothness assumption in the second implication cannot be omitted, as an easy construction (with B the convex hull of a Euclidean ball centred at the origin and two points x, x outside this ball) shows. Hug [139, 1.4] obtains a further result on surface area measures of bodies of constant Minkowski width, and he also gives an extension of this which is closely related to pairs of constant Minkowski width (for the latter notion see our Section 2.12 below). In [139] one can find also various related characterizations of unit balls. If a convex body K E 2 has sufficiently smooth boundary, one may denote its radius of curvature at x bd K with outward normal u = (cos α, sin α) by R(α). In this notation we can give a necessary criterion on a function R(α) to present the radius of curvature of a body K E 2 of constant width w(k) R + : R(α) + R(α + π) = w(k). (5) In Minkowski planes we have the same result. The sum of the radii of curvature at analogously opposite boundary points of a plane body of constant Minkowski width is constant and equals w(k). In terms of difference bodies (which are homothets of central symmetrals) this was first observed by Vincensini [295, p. 24], and Sz.-Nagy [281, p. 31], and Petty reproved this, see [216, Theorem (6.14)]. Chakerian [59] shows that for a figure K of constant width in M 2 the integral I(K) = 1 R 2 ds, 2 where R is the relative radius of curvature and ds denotes the relative arc length element, can also be expressed by the functional I(K) = n(x 1, x 2 )dx 1 dx 2, where n(x 1, x 2 ) denotes the number of diameters of K passing through x = (x 1, x 2 ) K. Let C be a twice continuously differentiable curve in E 2. The Four Vertex Theorem says that C has at least four vertices which are the points of C where the curvature of 11

12 this curve has a stationary value. Heil [129] derives a generalized version of the analogous theorem for Minkowski planes. His considerations imply that any curve of constant Minkowski width has at least six vertices. Defining the k-th quermassintegral of a convex body K E d in terms of mixed volumes by W k (K) := V (K,..., K, B,..., B) (here K appears n k times, and B appears k times, see [40, p. 49]), the equality W n k (K) = k ( ) k ( 1) i W n i (K)[w(K)] k i, k = 0, 1,..., n, (6) i i=0 holds if K is a body of constant width w(k) R +, cf. [77]. Chakerian [58] generalizes (6) as follows: For convex bodies K, L, M with K + L = M let V (K, k; ) denote the mixed volume with K occuring k times and representing n k fixed entries, and V (L, i; M, k i; ) be the mixed volume of i times L and k i times M with the same n k fixed entries as above. Then k ( ) k V (K, k; ) = ( 1) i V (L, i; M, k i; ). i i=0 Assuming L = K, M = w B (K) B and setting the remaining entries equal to the unit ball B of M d, an analogue of (6) for bodies of constant Minkowski width is obtained, see also [60]. Guggenheimer [124, p. 327] announces related results on bodies of constant width relative to a unit ball which is no longer centrally symmetric, i.e., only a gauge body. 2.8 Inequalities The Blaschke-Lebesgue Theorem states that among all figures of constant width w R + in E 2 only the Reuleaux triangle has minimum area (see [60, 7] and, for a short proof, e.g. [56]). The analogous theorem for Minkowski planes was obtained by D. Ohmann and, independently, by K. Günther in their dissertations (both in Marburg, 1948); Ohmann published his approach in [211], see also Petty [216, pp ]. Since Reuleaux triangles of fixed width in Minkowski planes can have different areas in general, we describe the construction of the particular type which this theorem refers to. Let the Minkowski unit vectors r 1, r 2, r 3 M 2 have the property r 1 + r 2 + r 3 = o. Then, by suitable translates of the segments [o, r i ], one can form a triangle a 1 a 2 a 3 with a 1 = o and a 2, a 3 bd B, cf. Figure 1. Connecting now the vertices a 1 and a 2 as well as a 2 and a 3 by corresponding boundary arcs of B (see again the figure), we get a figure of constant Minkowski width 1 whose boundary contains the points a 1, a 2 and a 3. Due to Ohmann any homothetical copy of such a figure is called a general Reuleaux triangle in M 2. With this notation we can formulate Theorem 11. Among all figures of constant width w R + general Reuleaux triangle has minimum area. in a Minkowski plane, a Chakerian [56] presents another proof of this theorem, and also Kubota and Hemmi [161] give an independent approach by investigating various inequalities for convex figures in E 2. From this they deduce that precisely the above mentioned analogues of Reuleaux triangles are extremal with respect to an inequality in terms of diameter and minimal 12

13 a 2 r 2 a 3 r 1 B r 3 o=a 1 Figure 1. Constructing a Reuleaux triangle width. Deriving a differential equation for the relative support function of a convex set, Ghandehari [103] gives an optimal control formulation of the Blaschke-Lebesgue theorem in Minkowski planes. The Firey-Sallee Theorem says that among all Euclidean Reuleaux polygons having n 3 vertices and width w R + the regular one has maximal area. In [238] Sallee announces without proof that his methods for getting this result in E 2 can be modified to obtain analogous results in Minkowski planes. Lenz [183] proves that in a Minkowski plane whose unit circle B is a Radon curve the ball has largest area among all bodies of fixed constant width. However, by the Rogers- Shephard inequality [236] in any Minkowski space the ball has the largest volume among all bodies of fixed constant width. Wernicke [300] shows that the area of a Reuleaux triangle of width 1 in a Minkowski, where equality holds iff the unit ball B is an affine regular hexagon (left hand side) or a parallelogram (right hand side). Furthermore, he proves that only in planes with B a parallelogram or a centrally symmetric hexagon there exist Reuleaux triangles that are Minkowski circles. Castro Feitosa [88] uses extremal properties of bodies of constant width in M 2 to extend various inequalities of Scott [246] for convex figures in E 2 to convex figures in Minkowski planes. E.g., he uses the facts that among all convex figures in M 2 with given diameter and circumradius the sets with largest thickness (= minimal width), perimeter, inradius or area are of constant width. Further inequalities for the area of figures of constant Minkowski width and related results are derived in [54] and [151]. In [151] an upper bound on the area of a figure C M 2 of constant width in terms of the Minkowski arc length of its pedal curve and other quantities is given; this bound is attained iff C is homothetic to a pedal curve of the isoperimetrix of M 2. Investigating products of dual cross section measures of convex bodies and their polar reciprocals, Ghandehari [104] proves that the d-dimensional volume of the polar reciprocal K of a body K M d of constant width 2 satisfies V (K ) V (B ), in which B is the polar reciprocal of the unit ball B. Here equality holds if and only if K = B. Another type of inequality is considered in [190]. For X a finite point set in E d with d- dimensional convex hull P, the points x i, x j X are called antipodal if there are different plane satisfies 1 6 Area( ) Area(B)

14 parallel supporting hyperplanes H, H of the polytope P with x i H, x j H. If E d is endowed with a Minkowski metric, one might ask for the number of pairs in X whose Minkowski distance is maximal. This number is not larger than the number a(x) of antipodal pairs in X, and if P is of constant Minkowski width, then equality holds. In [190] several upper bounds on a(x) are given. 2.9 Inscribed and circumscribed bodies As is well known, a hexagon (regular in the considered norm) can be inscribed in any Minkowski circle. We refer to [283, Chapter 4] for a broad and nice representation of related results and how this can be applied to construct curves of constant Minkowski width, in particular Reuleaux triangles. As a special case of a result of Doliwka [79] (conjectured by Lassak [172]) we have that any planar body of constant Minkowski width 1, say, has an inscribed pentagon whose vertices are in at least unit distance to each other. A generalization to arbitrary equilateral inscribed polygons is announced by Lassak (personal communication) Packings, coverings, lattice points Loomis [186] considers so-called three-coverings within the family of bodies of constant width w in Minkowski spaces. A set K three-covers the set L if L {a 1, a 2, a 3 } + K for some three points a 1, a 2, a 3. Among other results, Loomis shows that a Reuleaux triangle of width w three-covers any figure of constant width w. Further on, if B is a centrally symmetric octagon, then each figure of constant width three-covers every other set of the same constant width. Inspired by a question of P. C. Hammer, Sallee [237] considers bodies of constant width in association with lattices. In analogy to the Euclidean situation, he defines a Minkowski Reuleaux polygon to be a set of constant width w in M 2 which is the intersection of a finite number of (properly chosen) translates of wb. Saying that a set S avoids another set X if int S X =, he proves the following statements for any Minkowski plane with strictly convex unit ball B: Every set of maximal constant width avoiding a square unit lattice L is a Minkowski Reuleaux triangle P where each of the three open edges of P contains at least one point of L. If the lattice L is replaced by a locally finite family X of convex sets in an arbitrary Minkowski plane, then the corresponding maximal sets are Reuleaux polygons in M 2 all of whose open edges contain points from X. Surveys on the famous partition problem of Borsuk are presented by Grünbaum [122], [36, Chapter V] and Raigorodskii [229], see also [1, Chapter 15]. This problem is closely related to bodies of constant width, cf. [60], [131], [33] and also [36, Chapter VIII]. A first investigation of Borsuk s problem in Minkowski planes is due to Grünbaum [120]. He shows that if B is not a parallelogram, then any set of diameter 1 can be covered by three balls each of diameter less than 1. Let F M d be a bounded set of diameter h. What is the smallest integer k such that F is the union of k sets each of which has a diameter strictly smaller than h? Denoting this smallest number by a B (F ), Boltyanski and V. Soltan [39] prove that for d = 2 one has a B (F ) {2, 3, 4}, where a B (F ) = 4 occurs if and only if B is a parallelogram and the convex hull of F is a homothet of B. And a B (F ) > 2 holds if and only if one of the following two conditions is satisfied: (i) There is a unique completion of F to a figure C of constant width h. (ii) For any two parallel supporting hyperplanes of C at least one has nonempty intersection with F. 14

15 2.11 Rotors in polytopes It is obvious that bodies of constant width in E d are rotors in cubes and in some other polytopes. On the other hand, there are also polytopes with rotors not of constant width (but still having various similar properties). With this (a little extended) point of view, a remarkable list of related references can be taken from [40, pp ], [60, p. 80] and [131, p. 367]. Regarding Minkowski geometry, Ghandehari and O Neill [105] derive inequalities for the self-circumference U of rotors in equilateral triangles and figures of constant width. Here U is measured by taking these rotors or figures of constant width themselves as gauge figures or unit circles of a Minkowski plane. The inequalities compare their areas and mixed areas (taking also the polar reciprocal) with U. Some higher dimensional results are also given in [105]. Weakly related, Sorokin [275] studies certain classes of convex bodies which can roll in Minkowski spheres Concepts related to constant Minkowski width and further results Heil s concept of reducedness (cf. [130]) is in a sense dual to completeness. A convex body K E d which does not properly contain a convex body of the same minimal width is called a reduced body. For a discussion of results on reduced bodies see [131, 5.4]. The most striking open questions on reduced bodies in E d are: (a) Do there exist reduced polytopes for d 3? (b) Is a strictly convex reduced body in E d, d 3, necessarily of constant width? Recently Lassak and Martini [173] extended this notion to Minkowski spaces. It is easy to see that for certain norms (with the Manhattan norm as most simple case) and d 3 reduced polytopes exist, whereas in [173] the extension of (b) is again only verified for d = 2. Furthermore, it is shown that there exist reduced bodies in Minkowski spaces of dimensions d 3 having minimal width 1, say, but arbitrarily large diameter. Let δ(k, u) denote the (d 1)-dimensional volume of the orthogonal projection of a convex body K E d onto the subspace orthogonal to u S d 1. Usually δ(k, u) is called the brightness of K at u, cf. Chapters 3, 4, 8, and 9 of [98] for related results. A convex body K M d is said to be of constant brightness with respect to B if δ(k, u) is proportional to δ(b, u). Chakerian [57] shows that if both B and K in M 3 have C 2 boundary with everywhere positive curvature, then K is a homothet of B. More generally, Petty defines the Minkowski brightness of K M d at u S d 1 as the minimal Minkowski cross section area of the cylinder K +L, where L is the 1-subspace of direction u. In [217] and [218] he derives results on bodies of constant Minkowski brightness. The k-girth of a convex body K E d in direction u is given by the mixed volume dv (K,..., K, B,..., B, [u]), where K appears k times, the Euclidean ball B occurs n k 1 times, and [u] denotes the unit line segment parallel to u, see also Section 2.7 above. Chakerian [57] considers sets of constant k-girth in Minkowski spaces, and Petty [218] studies bodies of constant Minkowski curvature. Due to Maehara [188], two convex bodies K 1, K 2 E d are said to be a pair of constant width if K 1 + ( K 2 ) is a ball. Analogously, Sallee [241] defines a pair K 1, K 2 of convex 15

16 bodies to be a pair of constant width in M d if h(k 1, u) + h(k 2, u) = λ h(b, u) for some λ > 0 and all directions u S d 1. Among other results, he proves that K 1 is a summand of the unit ball B iff there is a K 2 such that {K 1, K 2 } is a pair of constant width in M d. Also Sallee s generalization of Theorem 5 with respect to sets having the spherical intersection property in M d (cf. Section 2.6 above) can be formulated in terms of pairs of constant width in M d, cf. [240]. Petty and Crotty [220] show that there are d-dimensional Minkowski spaces with convex bodies having exactly two equichordal points. Rodriguez Palacios [235] points out that results on summands of Banach spaces may be interpreted in terms of sets having constant width in Banach spaces. In view of multiplication with scalars, Minkowski addition and suitable combinations thereof, the family of all convex bodies in E d forms an abelian semigroup with scalar operators. Having such an algebraic structure and the Hausdorff metric in mind, Ewald and Shephard [87] introduce an equivalence class structure for the subclass of bodies of constant width, which yields an incomplete normed linear space. They remark that (due to a hint of Grünbaum) their respective results may be easily extended to bodies of constant Minkowski width. Such extensions were given by Sorokin [275], even for nonsymmetric unit balls which, on the other hand, have to be smooth and strictly convex. Taking the minimum width of certain representatives as a metric (in the above mentioned space), Lewis [185] shows that then a conjugate Banach space with complete norm is obtained. For related considerations we also refer to [86]. Finally we shortly mention the concept of bodies of constant affine width in the sense of affine differential geometry, see Süss [280] for an early contribution. For further references to this subject we refer to the final paragraphs of the surveys [60] and [131] and, in particular, to the investigations in [135] and [24], relating the concept of affine width to that of Minkowski width. 3 Generalized convexity notions in Minkowski geometry 3.1 Introduction In Section 3 we deal with modifications of the usual convexity notion, in most cases yielding natural extensions of basic theorems on convex sets. The main part will refer to a type of metric convexity which is usually called d-convexity, but also other kinds of convexity will be discussed. The letter d is used in two different meanings: for the dimension of the space, and for the historically fixed notions of d-segment and d-convexity. Since the distinction will always be clear from the context, we let it as it is. The notion of d-segment is based on so-called metric betweenness points and the metric betweenness relation which were first considered by Menger [206, Part I] and Blumenthal [27, Chapter II] in the context of complete, convex metric spaces. See also [207], [28, Chapter II], [114], [234], [5], [253], [29, Part 3], and [256]. Also based on betweenness points, Menger and Busemann proposed to complete Fréchet s axioms for a metric space to ensure the existence of geodesics, cf. [45, Chapters I and II], [47, Chapter I], and its continuations [48] and [51]. Replacing usual straight line segments in the common definition of convex sets by d-segments, the concept of d-convex sets is obtained, see [216], [114], [5], [53], [253], and [256] for the definition and first investigations. (We note that 16

17 Petty [216] speaks about the concept of Minkowski convexity.) Wider presentations of this concept can be found in the monographs [37], [270] and [36] (see the chapters with the headline d-convexity ), and it is also mentioned in the Handbook of Convex Geometry, cf. [193, 4]. For further generalized convexity notions we refer to [70, 9], [293] and [250]. 3.2 d-segments in Minkowski spaces Let γ be a simple curve in a d-dimensional Minkowski space M d which is parametrized by [t 0, t n ] and whose length is defined, in an elementary way, by γ := sup{ n a i 1 a i : n N, a i = γ(t i ), t 0 < t 1 < < t n }, i=1 with the endpoints a 0 = γ(t 0 ), a n = γ(t n ). A metric segment is a curve isometric to a closed segment of the real line, a metric line is a curve isometric to the real line, and a geodesic (cf. [45, 3] and [47, p. 32]) is a curve that is locally a metric segment, i.e., each point of the curve has a closed neighbourhood which is a metric segment. For a metric d(x, y) (= x y in a Minkowski space), the set [a, b] d := {z M d : d(a, b) = d(a, z) + d(z, b)} is called the d-segment with endpoints a, b M d. It is easy to show that any metric segment with endpoints a, b is contained in [a, b] d and, on the other hand, that [a, b] d is the union of all metric segments with endpoints a, b. Each point z M d satisfying a z b and d(a, b) = d(a, z) + d(z, b) is said to be a betweenness point of the distinct points a and b. See Menger [206, p. 77], who introduced betweenness points for arbitrary metric spaces and studied basic properties of the related betweenness relation (cf. also [27, Chapter II], and [28, Chapter II]). It is clear that the betweenness relation is closely related to the triangle inequality and therefore, in Minkowski spaces, to the (strict) convexity of the unit ball [109], cf. our first survey [196, 3]. In particular, any metric segment is a straight line segment iff the unit ball B of the Minkowski space is strictly convex, see [96], [73, p. 144], [42], [43], [257], and [76], also for further equivalent properties. We recall that strict convexity is equivalent to many notions such as the monotone property of the distance function and Chebyshev sets; see [196, 3] as well as the further references [9] and [92]. Extensions of this characteristic property to more general spaces and related observations are given in [252], [209], [42], [230], [94], [75], [95], [303], and [292]. See also [9] and [29, Chapters 6 and 7]. For detailed discussion of the following observations we refer to [196, 4]. Go l ab and Härlen [109] and, independently, Toranzos [290] have shown that the extreme points of the unit ball B of a Minkowski space coincide with the directions of unique metric segments, i.e., of curves γ from a to b such that γ = a b. Nitka and Wiatrowska [210] observe that the origin o is a betweenness point of p, q bd B iff the straight line segment [p, ϕ o (q)] bd B, where ϕ o (q) denotes the reflection of q at o. From this it follows that, given three arbitrary non-collinear points a, b, c M d, one can always find a norm such that b is betweenness point of a and c. To see this, it suffices to choose a Minkowski ball centred at b whose boundary contains [a, ϕ b (c)]. Verheul [294] says that a metric space X is modular if [a, b] d [b, c] d [c, a] d is nonempty for any triple a, b, c X, and X is called median if this intersection is always exactly one point. He shows that a Minkowski space is median iff it is isometric to l d 1, 17

18 the d-dimensional Minkowski space with norm x = d i=1 x i, i.e., iff the unit ball is a cross-polytope. Verheul uses lattice-theoretic techniques to study modular metrics and modular Banach spaces. Finally we want to give a geometric description of d-segments in terms of the boundary structure of the unit ball, see [257] and [36, 9]: For x, y M d, denote by B x, B y the Minkowski balls of radius x y with centres x and y, respectively. Furthermore, we write F x for the face of B x in bd B x that contains y and, analogously, F y for the face of B y in bd B y containing x, see Fig. 2. We denote by C x the cone with apex x consisting of all points x + λ(a + x), where a F x, λ 0, and by C y the cone with apex y and the representation y + λ(b y), b F y and λ 0. Then the d-segment with endpoints x, y is the intersection of the cones C x and C y (which are symmetric with respect to the midpoint of the straight line segment [x, y]), i.e., we have [x, y] d = C x C y. Soltan [270, Theorem 11.22] extends this observation to the infinite dimensional case. C x C y F x B y x [x, y] d y B x F y Figure 2. Construction of a d-segment 3.3 Fundamentals of d-convexity As Menger emphasizes in [206, Part I], the usual definition of convexity, using straight line segments, cannot be used for general metric spaces, and he defines a metric space X to be metrically convex if for any two distinct points x, y X there exists a betweenness point z, i.e., d(x, z) + d(z, y) = d(x, y) has to be satisfied. With this concept, sometimes also called Menger convexity or M-convexity (see, e.g., Busemann [47, 6]), convex metric spaces were studied, cf. the basic reference [28, 14] (or Section 3.8 below). A slight modification yields the notion of d-convexity which is more interesting for Minkowski spaces and seems to have been first defined by Petty [216] and, independently, by de Groot [114]. Using the term d-segment, one can formulate this definition (referring to a Minkowski space M d ) as follows. A set A M d is d-convex if for any points a, b A the d-segment [a, b] d is contained in A. Equivalently, A M d is d-convex provided for any three points a, b A, x M d, 18